Approximations of The Prime Soft Ideal and Maximal Soft Ideal of soft semirings

The notions of the fuzzy β -minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ , T -fuzzy rough set model ( FC ℐ T FRS ), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ , T -fuzzy rough sets ( CVP ℐ T FRS ). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FC ℐ T FRS based on fuzzy β -minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVP ℐ T FRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.

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Logical difference analysis based on variable precision lower approximation operator and graded upper approximation operator

2012 International Conference on Computer Science and Information Processing (CSIP) ◽

10.1109/csip.2012.6308987 ◽

2012 ◽

Author(s):

Xianyong Zhang ◽

Duoqian Miao

Keyword(s):

Approximation Operator ◽

Lower Approximation ◽

Difference Analysis ◽

Upper Approximation ◽

Variable Precision

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The Rough Linear Approximate Space and Soft Linear Space

Journal of Intelligent Systems ◽

10.1515/jisys-2014-0179 ◽

2016 ◽

Vol 25 (2) ◽

pp. 251-261

Author(s):

Yingcang Ma ◽

Shaoyang Li ◽

Yamei Liu

Keyword(s):

Boolean Algebra ◽

Linear Space ◽

Linear Algebra ◽

Rough Sets ◽

Linear Subspace ◽

Real Life ◽

Inclusion Relation ◽

Soft Sets ◽

Lower Approximation ◽

Upper Approximation

AbstractThe studies of rough sets and soft sets, which can deal with uncertain problems in real life, have developed rapidly in recent years. We have known that linear space is a very important concept in linear algebra, so the aim of this paper was mainly focused on combining research in linear space, rough sets, and soft sets. First, according to the properties of upper (lower) approximation in rough linear space, the inclusion relation of the upper approximation’s union and the inclusion relation of the lower approximation’s intersection are improved. The equations of the upper approximation’s union and the lower approximation’s intersection are given. Secondly, the connection of linear space to rough sets is explored and the rough linear approximate space is proposed, which is proved to be a Boolean algebra under the intersection, union, and complementary operators. Thirdly, the combination of linear space and soft set is discussed, the definitions of soft linear space and soft linear subspace are proposed, and their properties are explored. Finally, the definitions of lower and upper approximation of a subspace X in soft linear space are given and their properties are studied. These investigations would enrich the studies of linear space, soft sets, and rough sets.

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Non Homogeneous Rough Finite State Automaton

Revista Gestão Inovação e Tecnologias ◽

10.47059/revistageintec.v11i2.1700 ◽

2021 ◽

Vol 11 (2) ◽

pp. 629-641

Author(s):

B. Praba ◽

R. Saranya

Keyword(s):

Artificial Intelligence ◽

Machine Learning ◽

Dynamical Behaviour ◽

Finite State Automaton ◽

Finite State Automata ◽

Lower Approximation ◽

Upper Approximation ◽

Finite State ◽

Transition Map

Objective: The study of finite state automaton is an essential tool in machine learning and artificial intelligence. The class of rough finite state automaton captures the uncertainty using the rough transition map. The need to generalize this concept arises to adhere the dynamical behaviour of the system. Hence this paper focuses on defining non-homogeneous rough finite state automaton. Methodology: With the aid of Rough finite state automata we define the concept of non-homogeneous rough finite state automata. Findings: Non homogeneous Rough Finite State Automata (NRFSA) Mt is defined by a tuple (Q,Σ,δt,q0 (t),F(t)) The dynamical behaviour of any system can be expressed in terms of an information system at time t. This leads us to define non-homogeneous rough finite state automaton. For each time ‘t’ we generate lower approximation rough finite state automaton Mt_ and the upper approximation rough finite state automaton Mt- and the defined concepts are elaborated with suitable examples. The ordered pair , Mt=(M(t)-,M(t)-) is called as the non-homogeneous rough finite state automaton. Conclusion: Over all our study reveals the characterization of the system which changes its behaviour dynamically over a time ‘t’. Novelty: The novelty of the proposed article is that it clearly immense the system behaviour over a time ‘t’. Using this concept the possible and the definite transitions in the system can be calculated in any given time ‘t’.

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The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Complexity ◽

10.1155/2021/5523822 ◽

2021 ◽

Vol 2021 ◽

pp. 1-10

Author(s):

Qiao-Ling Song ◽

Hu Zhao ◽

Juan-Juan Zhang ◽

A. A. Ramadan ◽

Hong-Ying Zhang ◽

...

Keyword(s):

Complete Lattice ◽

Lattice Isomorphism ◽

Double Negative ◽

Lattice Structures ◽

Fuzzy Relations ◽

Lower Approximation ◽

Upper Approximation ◽

Approximation Operators ◽

Neighborhood System ◽

Generalized Neighborhood System

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism.

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Error bounds for approximations with deep ReLU neural networks in Ws,p norms

Analysis and Applications ◽

10.1142/s0219530519410021 ◽

2019 ◽

Vol 18 (05) ◽

pp. 803-859 ◽

Cited By ~ 3

Author(s):

Ingo Gühring ◽

Gitta Kutyniok ◽

Philipp Petersen

Keyword(s):

Neural Networks ◽

Approximation Error ◽

Upper And Lower Bounds ◽

Trade Off ◽

Lower Approximation ◽

Upper Approximation ◽

Regular Functions ◽

Function Classes ◽

The Neural Network ◽

Sobolev Norms

We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently approximate Sobolev regular functions if the approximation error is measured with respect to weaker Sobolev norms. In this context, we first establish upper approximation bounds by ReLU neural networks for Sobolev regular functions by explicitly constructing the approximate ReLU neural networks. Then, we establish lower approximation bounds for the same type of function classes. A trade-off between the regularity used in the approximation norm and the complexity of the neural network can be observed in upper and lower bounds. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

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Generalized Rough Sets

Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing - Advances in Computational Intelligence and Robotics ◽

10.4018/978-1-4666-9798-0.ch011 ◽

2016 ◽

pp. 223-246

Author(s):

Mona Hosny ◽

Ali Kandil ◽

Osama A. El-Tantawy ◽

Sobhy A. El-Sheikh

Keyword(s):

Rough Set ◽

Rough Set Theory ◽

Order Relation ◽

Boundary Region ◽

Current Approach ◽

Partially Order ◽

Lower Approximation ◽

Upper Approximation ◽

Relation Ideal ◽

Lower And Upper Approximations

This chapter concerns construction of a new rough set structure for an ideal ordered topological spaces and ordered topological filters. The approximation space approached depend on general binary relation, partially order relation, ideal and filter concepts. Properties of lower and upper approximation are extended to an ideal order topological approximation spaces. The main aim of the rough set theory is reducing the bouwndary region by increasing the lower approximation and decreasing the upper approximation. So, in this chapter different methods are proposed to reduce the boundary region. Comparisons between the current approximations and the previous approximations (El-Shafei et al.,2013) are introduced. It's therefore shown that the current approximations are more generally and reduce the boundary region by increasing the lower approximation and decreasing the upper approximation. The lower and upper approximations satisfy some properties in analogue of Pawlak's spaces (Pawlak, 1982). Moreover, we give several examples for comparison between the current approach and (El-Shafei et al., 2013).

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